We propose a modelling framework to analyse the stochastic behaviour of heterogeneous, multi-scale cellular populations. We illustrate our methodology with a particular example in which we study a population with an oxygen-regulated proliferation rate. Our formulation is based on an age-dependent stochastic process. Cells within the population are characterised by their age (i.e. time elapsed since they were born). The age-dependent (oxygen-regulated) birth rate is given by a stochastic model of oxygen-dependent cell cycle progression. Once the birth rate is determined, we formulate an age-dependent birth-and-death process, which dictates the time evolution of the cell population. The population is under a feedback loop which controls its steady state size (carrying capacity): cells consume oxygen which in turn fuels cell proliferation. We show that our stochastic model of cell cycle progression allows for heterogeneity within the cell population induced by stochastic effects. Such heterogeneous behaviour is reflected in variations in the proliferation rate. Within this set-up, we have established three main results. First, we have shown that the age to the G1/S transition, which essentially determines the birth rate, exhibits a remarkably simple scaling behaviour. Besides the fact that this simple behaviour emerges from a rather complex model, this allows for a huge simplification of our numerical methodology. A further result is the observation that heterogeneous populations undergo an internal process of quasi-neutral competition. Finally, we investigated the effects of cell-cycle-phase dependent therapies (such as radiation therapy) on heterogeneous populations. In particular, we have studied the case in which the population contains a quiescent sub-population. Our mean-field analysis and numerical simulations confirm that, if the survival fraction of the therapy is too high, rescue of the quiescent population occurs. This gives rise to emergence of resistance to therapy since the rescued population is less sensitive to therapy.

f0025: Plot showing how the G1/S transition age, , changes as the ratio , which is determined by the ratio of the (conserved) amounts of SCF activating and inactivating enzymes (Alarcón, 2014, de la Cruz et al., 2015), varies. We show for c=1 (blue circles) and c=0.1 (red squares). Parameter values as given in Table B2. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Mentions:
With this in mind, we can analyse the effect of changing the relative concentration of SCF-activating and inactivating enzymes on the timing of the G1/S transition. Our results are shown in Fig. 4, Fig. 5. Fig. 4 illustrates that, for a fixed oxygen concentration, the G1/S transition is delayed by depriving the system of SCF-activating enzyme: as the ratio of SCF activating and deactivating enzyme increases, the G1/S transition takes longer to occur. Then, Fig. 5 shows that the G1/S transition age decreases when increases. Furthermore, increasing the oxygen concentration c from c=0.1 to c=1 shifts the curve towards lower transition ages . Note that this prediction is beyond the reach of the mean-field limit Bedessem and Stephanou (2014).

f0025: Plot showing how the G1/S transition age, , changes as the ratio , which is determined by the ratio of the (conserved) amounts of SCF activating and inactivating enzymes (Alarcón, 2014, de la Cruz et al., 2015), varies. We show for c=1 (blue circles) and c=0.1 (red squares). Parameter values as given in Table B2. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Mentions:
With this in mind, we can analyse the effect of changing the relative concentration of SCF-activating and inactivating enzymes on the timing of the G1/S transition. Our results are shown in Fig. 4, Fig. 5. Fig. 4 illustrates that, for a fixed oxygen concentration, the G1/S transition is delayed by depriving the system of SCF-activating enzyme: as the ratio of SCF activating and deactivating enzyme increases, the G1/S transition takes longer to occur. Then, Fig. 5 shows that the G1/S transition age decreases when increases. Furthermore, increasing the oxygen concentration c from c=0.1 to c=1 shifts the curve towards lower transition ages . Note that this prediction is beyond the reach of the mean-field limit Bedessem and Stephanou (2014).

We propose a modelling framework to analyse the stochastic behaviour of heterogeneous, multi-scale cellular populations. We illustrate our methodology with a particular example in which we study a population with an oxygen-regulated proliferation rate. Our formulation is based on an age-dependent stochastic process. Cells within the population are characterised by their age (i.e. time elapsed since they were born). The age-dependent (oxygen-regulated) birth rate is given by a stochastic model of oxygen-dependent cell cycle progression. Once the birth rate is determined, we formulate an age-dependent birth-and-death process, which dictates the time evolution of the cell population. The population is under a feedback loop which controls its steady state size (carrying capacity): cells consume oxygen which in turn fuels cell proliferation. We show that our stochastic model of cell cycle progression allows for heterogeneity within the cell population induced by stochastic effects. Such heterogeneous behaviour is reflected in variations in the proliferation rate. Within this set-up, we have established three main results. First, we have shown that the age to the G1/S transition, which essentially determines the birth rate, exhibits a remarkably simple scaling behaviour. Besides the fact that this simple behaviour emerges from a rather complex model, this allows for a huge simplification of our numerical methodology. A further result is the observation that heterogeneous populations undergo an internal process of quasi-neutral competition. Finally, we investigated the effects of cell-cycle-phase dependent therapies (such as radiation therapy) on heterogeneous populations. In particular, we have studied the case in which the population contains a quiescent sub-population. Our mean-field analysis and numerical simulations confirm that, if the survival fraction of the therapy is too high, rescue of the quiescent population occurs. This gives rise to emergence of resistance to therapy since the rescued population is less sensitive to therapy.